Lexington, MA SCRABBLE® Club Rating System

This long page explains the workings of the Lexington Club
Rating System. It has been in effect since early 1981 and
has been a successful indicator of the relative strengths of the
players. The system was designed primarily by Alan Frank, with
assistance from a few others, including: Steve Root, Mike
Wolfberg, Eric Albert, and Frank Voss. Mike authored the computer
program which does the job of supporting this system. That
program has undergone changes a few times, especially as it has
undergone changes in the programming language used, but the basic
mathematics of this rating system have remained unchanged.

The basic idea of the system is that each player acquires a
numerical rating which is an indication of the player's relative
strength in the club. The rating numbers have no intrinsic
meaning; they are just meaningful in how they relate to the other
ratings. The system was set up to have the top player rated
around 1800; we expected the lowest rated players would still be
above 1000, and so a low-looking 3-digit number would not be
used.

Unlike the national rating system, in which only win/loss is
important, the Lexington Club Rating System does take into
account game scores. Rather than focussing on point spreads or
pure scores, the quantity of importance is the percentage of
total points a player scores. Thus a point spread of 10 points is
considered more important in a low-scoring game than in a
high-scoring game. In the computations, the winner's percentage
of total points is boosted by 4% (and the loser's percentage is
reduced by 4%) to give a premium to the win. The boost is not
used in a tied game.

When a player first shows up at the club, during the first ten
games, the rating a player gets is based on performance against
the other players, who, for the most part, have established
ratings. The new player's conditional rating is based on the game
scores in relation to that player's opponents. For the purpose of
computing the new player's opponents' ratings, the new player is
assigned a guessed initial rating by the rating statistician,
based on initial performance. This is typically 25-50 rating
points higher than the first night's numbers say. This has the
effect of pumping more points into the system when new players
show up. It also gives the opponents of new players a small
advantage in getting their own ratings increased, since the
assigned rating may be a bit higher than the real level of the
new player. On the other hand, a serious new player will likely
increase in strength even over the first ten games played in the
club, so the extra boost is also a prediction.

The mathematics for a new player are sufficiently complex that
this presentation will skip this. If there is some demand for
details on this subject, its presentation will be included in the
future. Let us go on to explain the workings of the rating system
for those players who have played at least 10 games at the club,
as remembered in the club data. Each player is remembered in the
club data for up to 99 weeks of inactivity. Once that time has
passed, a returning player starts all over in the developing of
statistics.

The club operates on a fiscal year which restarts each
September. For the year (from September through August) each
player's number of wins, number of losses, and average scores are
computed. As the new season begins, these are restarted afresh,
but the ratings are carried across the season boundary. An
exception to this is when all ratings of the club members are
boosted. This has been done only once since the inception of the
system. It can easily be done again at a season boundary if and
when we notice a general degradation of the ratings. The clue for
this will be the observation that the highest rated player is far
below 1800.

The idea of the rating system is that any two players in the
club can play each other - even the strongest and the weakest,
and the system provides a kind of handicapping mechanism. Based
on the difference between the two players' ratings, there are
expected outcomes of the games. For example, if the highest and
lowest rated players were to play on 23-Jul-98 and have a game in
which the higher-rated winner gets 433 and the opponent gets 286,
then that is below par for the winner. For that game total, the
score is predicted to be 470-249. The winning percentage should
be 69.3% (including the 4% boost), but the game scores indicate
the winner got 64.2% (boost included). This causes the winner to
lose 3 rating points and the opponent to gain 3 rating points.
Some real-life examples are given later on this page.

The following table indicates par game scores for various
rating differences between the two players. For example, if
strong player (with rating 1805) plays a weaker player (with
rating 1605), the rating difference is 200. An expected score may
be 405-295 (when their total points are 700). If this or some
other game scores on the 200 row was achieved,
their ratings do not change.

TOTAL POINTS SCORED

RATING
DIFF.

450

500

550

600

650

700

750

800

850

900

0-37

225-225

250-250

275-275

300-300

325-325

350-350

375-375

400-400

425-425

450-450

40

226-224

252-248

277-273

302-298

327-323

352-348

377-373

402-398

428-422

453-447

50

230-220

255-245

281-269

306-294

332-318

357-343

383-367

408-392

434-416

459-441

75

236-214

263-237

289-261

315-285

341-309

368-332

394-356

420-380

446-404

473-427

100

242-208

269-231

296-254

323-277

350-300

377-323

404-346

430-370

457-393

484-416

125

247-203

275-225

302-248

330-270

357-293

385-315

412-338

440-360

467-383

495-405

150

252-198

280-220

308-242

336-264

364-286

392-308

420-330

448-352

476-374

504-396

175

256-194

285-215

313-237

342-258

370-280

399-301

427-323

456-344

484-366

513-387

200

260-190

289-211

318-232

347-253

376-274

405-295

434-316

463-337

492-358

521-379

225

264-186

294-206

323-227

352-248

382-268

411-289

440-310

470-330

499-351

528-372

250

268-182

298-202

327-223

357-243

387-263

417-283

446-304

476-324

506-344

536-364

275

271-179

301-199

331-219

362-238

392-258

422-278

452-298

482-318

512-338

542-358

300

275-175

305-195

336-214

366-234

397-253

427-273

458-292

488-312

519-331

549-351

325

278-172

309-191

339-211

370-230

401-249

432-268

463-287

494-306

524-326

555-345

350

281-169

312-188

343-207

374-226

405-245

437-263

468-282

499-301

530-320

561-339

375

284-166

315-185

347-203

378-222

410-240

441-259

473-277

504-296

536-314

567-333

400

286-164

318-182

350-200

382-218

414-236

446-254

477-273

509-291

541-309

573-327

425

289-161

321-179

353-197

386-214

418-232

450-250

482-268

514-286

546-304

578-322

450

292-158

324-176

357-193

389-211

422-228

454-246

486-264

519-281

551-299

584-316

475

294-156

327-173

360-190

393-207

425-225

458-242

491-259

523-277

556-294

589-311

500

297-153

330-170

363-187

396-204

429-221

462-238

495-255

528-272

561-289

594-306

525

299-151

333-167

366-184

399-201

433-217

466-234

499-251

532-268

566-284

599-301

550

302-148

335-165

369-181

403-197

436-214

470-230

503-247

537-263

570-280

604-296

575

304-146

338-162

372-178

406-194

439-211

473-227

507-243

541-259

575-275

608-292

600

307-143

341-159

375-175

409-191

443-207

477-223

511-239

545-255

579-271

613-287

625

309-141

343-157

377-173

412-188

446-204

480-220

515-235

549-251

583-267

618-282

650

311-139

346-154

380-170

415-185

449-201

484-216

518-232

553-247

587-263

622-278

675

313-137

348-152

383-167

418-182

452-198

487-213

522-228

557-243

592-258

626-274

700

315-135

350-150

385-165

420-180

455-195

491-209

526-224

561-239

596-254

631-269

When two rated players have played, this is how the rating
change is computed. First, compute the quantity WINNER-PERCENT,
as:

WINNER-PERCENT = 100 * WINNER-SCORE / (WINNER-SCORE + LOSER-SCORE)

but then it is boosted by 4 when the game is not tied, so

if (WINNER-SCORE is not equal to LOSER-SCORE) then
WINNER-PERCENT = WINNER-PERCENT + 4.0

Then compute the expected percentage difference by this
formula (originally determined by looking at game data):

The "log" in the above computation is the natural logarithm,
sometimes denoted as "ln", such as in the MS Windows Calculator program.
It is named "log" in the C programming language run-time library.

The following table presents the full rating change as a
result of a game given the percentage difference between the
expected and actual percentages.

PERCENT
DIFF.

RATING
CHANGE

0

-

12

same

13

-

14

13

15

14

16

-

17

15

18

-

19

16

20

-

21

17

22

-

23

18

24

-

25

19

26

-

28

20

29

-

31

21

32

-

34

22

35

-

38

23

39

-

42

24

43

-

46

25

47

-

51

26

52

-

57

27

58

-

63

28

64

-

70

29

71

-

77

30

78

-

85

31

86

-

94

32

>

94

33

The winner is boosted by RATING-CHANGE (rounded to the nearest
integer) when that player has played less than 50 games at the
club; otherwise, the winner is boosted half of that (rounded to
the nearest integer). The idea is that until a player is
established in the club's records, that player's rating changes
more quickly. It is assumed veteran players' ratings deserve to
change more slowly.

Similarly, the loser is reduced by either RATING-CHANGE or
half of RATING-CHANGE, depending on the number of games the loser
has played.

Once a player has played at least ten games in the club, the
above rules apply. The old ratings used in the computations are
the ones published on the news sheet, so rating changes for each
game are accumulated for the entire session before they are used
to update the current data. This implies the order of the
recorded games does not matter (for players who have played at
least 10 games).

Here are a few examples of how the system works. They are
based on real data representing games played 23-Jul-98. Let us
say these folks have played more than 50 games in the club and
have these ratings:

PLAYER

RATING

A

1824

B

1805

C

1713

D

1708

E

1610

F

1588

Now, here are games they played. The "PAR GAME"
column indicates the expected game score when the total points
scored were the same as the actual total. The presented
percentages include the 4% adjustments to the players. You can
predict the rating change by computing the difference in the
percentages. For example, in the first game, the difference is
approximately 8. For numbers in this range, the change in rating
is the same as the percentage difference. Half the rating change
is used for players who have played more than 50 games since they
began at the club, you see a rating change of 4 (half of 8).
Rating changes are made in whole numbers.

WINNER

LOSER

PAR
GAME

EXPECTED
WINNER
PERCENT

ACTUAL
WINNER
PERCENT

WINNER
RATING
CHANGE

LOSER
RATING
CHANGE

A

459

D

272

399-332

58.6%

66.8%

+ 4

- 4

C

440

A

399

383-456

41.7%

56.4%

+ 7

- 7

A

429

E

325

440-314

62.3%

60.9%

- 1

+ 1

D

424

E

314

396-342

57.7%

61.5%

+ 2

- 2

C

512

E

267

420-359

58.0%

69.7%

+ 6

- 6

A

421

E

236

383-274

62.3%

68.1%

+ 3

- 3

C

354

B

326

317-363

42.6%

56.1%

+ 7

- 7

B

419

F

297

418-298

62.4%

62.5%

0

0

You can find out the rating changes by providing the game score
and players' ratings by using the
Lexington SCRABBLE® Club Rating Changes Calculator
here on this web site. There is a similar program (named DeltaRat)
on the laptop PC usually operating at club sessions.